2977
Derivation of the Thermodynamics of Polymer Solutions
A Derivation of the Thermodynamics of Polymer Solutions hvsugh Use of the Free
Volume Concept. B. The Heat of Mixing Jean Dayantis C.N. R.S., Centre de Recherches sur les Macromolecules, 67-Strasbourg, France
(Received February 72, 7973)
Publication costs assisted by the Centre de Recherches sur les Macromolecules
Theoretical relationships for the enthalpy of mixing polymer with solvent have been derived using the free volume concept. The theory is not restricted to polymer solutions but should apply to mixtures of simple liquids as well. To calculate the effect of the “expansion” of the polymer and of the “compression’’ of the solvent which occurs during the mixing process, following the scheme adopted in this work, the function W ( T ) = a(T)y(T)VM(T)T has been used where cy is the thermal expansion coefficient, y the thermal pressure coefficient, V,, the molar volume, and T the absolute temperature. The theory has been developed assuming first that the mixing free volume fractions of the solvent clrn and of the polymer are equal to the free volume fraction urn of the mixture. Then, the theory has been modified in order to include the case where u l r n # uzm # u m. The following n-alkane mixtures have been investigated: 68-c16, C6-C16, c6-c22, and c6-c36. Also polyisobutylene mixtures with hexane, octane, and hexadecane have been investigated. It is found theoretically, upon assuming that i l l r n = u p = grn, that a t room temperature c8-Cl6, c6-c16, and the polyisobutylene-n-alkane mixtures should have small and positive pair interchange energies. This is in conformity with experience. However, as the temperature is increased from 20 to loo”, this interchange energy is found to become negative for C!3-c16 and c6-cl6 mixtures, unless the hypothesis of equal free volume fractions in the pure components and in the solution is abandoned. Other mixtures than CS-cl6 and c6-Cl6 mixtures could not be presently investigated a t higher temperatures due to the lack of data for a and especially for y as a function of temperature.
I. Introduction In a preceding article’ the entropy of mixing of polymer with solvent has been derived using the free volume concept. The present paper is concerned with the enthalpy of mixing derived on the same basis. Only nondilute solutions are considered, This article, as well as the previous one. is conceived as an attempt toward deriving the thermodynamics of solutions and especially polymer solutions independently of any cell model theory of the liquid state.2-10 Since the cell model theory of the liquid state is only an idealized and approximate description of real liquids, the idea of deriving the thermodynamics of solutions using the free volume concept, but independently of any cell model theory, may be a useful one, and deserves a t least to be carefully explored. Thorough critical reviews of cell model theories of the liquid state have been given by Barker1’ and by Levelt and Cohen.12 Consideration of these reviews leads to the conclusion that the assumed similarity of the liquid and the solid structure, although useful in some cases, does not generally lead to relevant values for the parameters of the liquid state ( e . g . , the coefficient of thermal expansion a or the coefficient of isothermal compressibility K ) . These deficiencies should be even more apparent when solutions are considered. as shown previously. The free volume in a pure component is defined as the volume @ ( T )at the temperature T”K minus the volume @ ( O ) a t 0°K. A more relevant term for this quantity would be the “expansion volume,” the term free volume being reserved for the empty volume, as pointed out by Haward in his recent review on occupied v01umes.l~The term free volume will be used instead throughout this work, since it is the one most often used. It appears that the free vol-
ume, as above defined, is the relevant physical quantity to use when adopting a free volume approach t o the problem of solutions. The free volume fraction L‘,of component i is taken equal to
These definitions being recalled, the mixing process will now be assumed to be equivalent to the following scheme.’ (a) The solvent is compressed from the initial volume @lo to the mixing volume whereas the polymer is expanded from the initial volume @lo to the mixing volume !t?2m
=
@Zn(l
- u*)/(l - urn)
where L‘m
= VlUl
+
C21’2
(3)
(4)
and p z being the volume fractions of solvent and of polymer. (b) Compressed solvent and expanded polymer are mixed without volume change. (c) The mixture is compressed (or expanded) to its true value so as to take into account the excess volume of mixing. Let us recall that Scott14 and McGlashan, et u L . , ’ ~ have considered various mixing processes. The process considered by these authors which approaches most the above scheme is the so called “equal volume per segment” mixing process. The reader is refered to the original papers for further comparison. p1
The Journal of Physical Chemistry, Vol. 77, No. 25. 7973
2978
Jean Bayantis
The above scheme for the mixing process, if relevant, should apply to all mixtures of nonelectrolytes, whether mixtures of simple liquids or polymer-solvent mixtures. For usual simple liquids a t room temperature and atmospheric pressure, the free volume fraction varies within a rather narrow range of values. Consequently, the free volume fraction in the mixture should be close to that in the pure components and effects arising from free volume fraction differences should be generally small. Therefore in this case, as expected, the regular solution theory, using a rigid lattice as background, constitutes a quite acceptable approximation in many instances. On the other hand, an obvious fact when polymer-solvent mixtures are considered is that in the pure components the free volume fractions are quite different. A value of p = u z / u l near one-half has been estimated in the preceding paper. Haward's review on occupied volumes supports this estimation.13 Now, in the mixture, the free volume fraction urn will have some intermediate value between u1 and UZ. As a result of this and of thermal agitation, it is quite impossible that the mean distance between solvent-solvent pairs in the mixture will retain the same value uTlo as in the pure solvent. An equivalent argument may be considered for the polymer-polymer segment pairs, so that we generally expect that
(5) and
(6)
heated (or cooled) a t constant pressure until the volume reaches the required value @p; (2) the component may be compressed (or depressed) isothermally and reversibly until the volume reaches the value a L m . Process 1 leads to an expression depending on parameters which are readily accessible experimentally and will be the only one to be considered here. Only the work of the internal pressure contributes in step a of the mixing process to the heat of mixing. T o calculate the work of the internal pressure (aE/dV),, one may multiply both sides of the so-called thermodynamic equation of state by d V
P,, dV
= T(dP/dT)v
(11) If the external pressure P,, is sufficiently small so that the elementary work P,, d V may be neglected, then one may write dQ = P,, dV = T(dP/dT)v dV (12) As previously indicated we shall consider that the volume changes by varying the temperature, while the pressure is kept constant. Introducing the coefficient of thermal expansion eq 12 becomes, considering component i dQt = ~,(T)T,(T)V/M'(T)T dT (12') where uL is the coefficient of thermal expansion, y L the thermal pressure coefficient, Vlf1 the molar volume of component i, and T the absolute temperature. The total heat evolved during the isobaric compression and expansion processes will be equal to
The simplest possible assumption is that -
-
-
all = aZ2= alz (7) and this leads to the previous scheme for the mixing process. More generally, when the simplest assumption, eq 7, does not hold, one should write, considering inequalities 5 and 6
~
-
-
a220 aZ2< aI2 < all u1
u l m > Urn
d V - (dE/aV)T d V
II
=
I,
=
L T0 o + A T I W l (dT T)
(13a)
T,+AT?
W k T ) dT
(13b)
'1
where T.'Vl(T)
~ , ( T ) ? , ( T ) V M ' ( TdT ? i
1.2
(14)
-
(loa)
(lob)
and
< vqm < u,
(10c) In order that the scheme assumed for the mixing process be of any use, the contributions due to steps a, b, and c must be evaluated and this is done in section 11. L'2
11. Heat o f Mixing
In a first approach the simplest assumption 7 will be assumed to hold and the corresponding relationships derived. In a second stage, these relationships will be corrected so as to take into account inequalities 10. Step a. First the heat contributions arising from the compression of the solvent from the volume @loto the volume @lrnand from the expansion of the polymer from will be evaluated. There the volume azoto the volume @ p are two differentways of bringing component i from the volume aL0to the volume a Z m : (1) the component may be The Journal of Physicai Chemistry, Voi. 77, No. 2 5 , 7973
urn
= PlUl
+
(17)
p2L'z
(yl and are mean values of ocl(T)and m ( T ) in the temAT1 and To, TO A T z . If now perature ranges To, To we mix n1 moles of solvent with n2 moles of polymer, the temperature being To and the pressure sufficiently low, the heat exchange with the surroundings due to the work of the internal pressures of the solvent and of the polymer will be equal to
+
+
AQinternalpress =
nlIl
+ n2I2=
n, fT"iT'Wl(T) J
To
dT
+ n2 f
Tc+AT?
W2(T) dT (18)
J7,
This quantity is readily determined if the three coefficients CY, y, and Vxf, all of them being readily accessible experimentally, are known as a function of temperature. 1
2979
Derivation of t h e Thermodynamics of Polymer Solutions
Step b. This is the usual interchange energy contribution, the only one taken into account in Flory's old theory of solutions. Let us write =
Qinterch energq
where
__
Awiz
(20)
+ G2) -
= '/?(GI
1, + nl(dll/dnl) =
Ah1
n2(dI2/anl)
+-
( ~ 2 ~ 1 3
+q + Ah1
Ah1
+ (n, + nz>(aI,/an,?
1 4
(19)
RTr'hPz
{ = zAwlz/hT -
a
(30)
A T Using the two eq 13 and eq 24 for 11,4, and 1, the above expression is found equal to
~
1
2
(21)
2 i i G is the mean energy change when replacing one solvent-solvent and one polymer-polymer pair by two polymer-solvent pairs, z being the number of nearest neighbors. { is therefore the original Flory x. Letting
kT{
-
z A w ~= ~13
(22)
one obtains (23) Step c. The heat contribution arising from the isothermal compression (or expansion) of the mixture from the volume with no excess volume to the real volume will be set equal to Qinterchenergy
= nl{z13
(nl + ni?)WdT~+ AT3)(a(AT3>/anl> (30') AT1, iiT2, and AT3 are given by eq 15, 16, and 26, respectively. d(ATl)/anl and a(AT2)/dnl are easily derived from eq 15 and 16 and one obtains
-a(AT1) - _ -an1
1 n1
~
-
- uJ(1 - u ) )
1 ( ~~ 1 2
(1 -
a1
(31)
u,)2
In order to obtain an expression for d(AT3)/an1some hypothesis has to be made regarding the variation of @E with the volume fraction p L of one of the components. It will be assumed that a fair description of the excess volume is generally given by the equation
aE/'(ni+ n2> = A d 1 - (FJ
(33)
whereA is an experimental constant. Then, from
using eq 33 and the fact that by hypothesis, step b cf, =
It is thus assumed that by taking arithmetic mean values for a12, y l 2 , and Vh1l2 only a negligible error in introduced. The excess volume of mixing will be considered presently to be an additional parameter which will have to be experimentally determined. However, will be determined within the framework of the present theory in a future article. Summing up contributions a, b, and c, one obtains for the enthalpy of mixing at low pressure nl moles of solvent and n2 moles of polymer
AH
=
AQinteinalpress AQinierchenergy n111 + 7 2 2 1 2 + n1cp21,
+ AQexceesooi + (nl + nJ1, (28)
The experimental determination of AH requires thus the knowledge of a , -y, and V Mas a function of temperature for both components, of 13 = zAwl2, and, also, of the molar excess volume + M E = +E/(nl n2) as a function of
+
(n2. 111. Partial Molar Heats of Mixing. a. Partial Molar Heat of Mixing of the Solvent. By definition
G = [ ~ A H / ~ ~ I I T , P , ~ ~ (29) From eq 28 one immediately obtains
+ n2Vh?
nlVd
one obtains, Vh,1'2being the molar volume of 1 mol of segments (ni n d l - ~ 2 ) f i
+
T,P,n2
nlVd
a12
+ n2Vhf2
is thus entirely determined using the same parameters as those necessary to determine AH. b. Partial Molar Heat of Mixing of the Polymer. Since
ilhe = [dAH/dn&,~,,,
(35)
one obtains, from eq 28
-
Ah2 =
(dIl/dnJ Ah2
12
+ n2(d12/anJ
+-+y+
Ah2'
I,
+
Ah23 n2(dI,/dn2)
Ah24
(36)
The explicit form of this equation may be evaluated as for the partial molar enthalpy of the solvent using eq 13 and 24.
IV. Flory-Huggins Interaction Parameter X. As pointed out by Guggenheim16 many years ago, the Flory-Huggins interaction parameter x should be written The Journal of Physical Chemistry, Vol. 77, No. 25, 1973
2980
Jean Dayantis
as the sum of an entropic and of an enthalpic contribution X = X S f XH (37) Using the free volume concept, the entropic part xS of x has already being derived in the previous paper, yielding
where it is recalled that p is the ratio u z / u l and c' = w ( c / ~ ) ( V ~ I ~ / VcM being ' ~ ) ,the Prigogine parameter for the external degrees of freedom of the polymer segments and w a constant lying being and %. On the other hand, the enthalpic part X H is given by
and
dT
(13b')
where
In eq 15' and 16' inequalities 10 for u l m and uzm have to be fulfilled. It is convenient to introduce two constants K1 and K z such that T,+AT,'
I,'
=
[
W,(T) dT
=
JTO
or XH =
[AT ,
+ ah,2 + nh,3 + ah141
(40)
To+AT,
a
where the values of the A T are indicated in eq 30 and 30'. The total x thus appears as the sum of xs and xH given by eq 38 and 40, respectively. The explicit expression is too cumbersome to be written on a single line. Nevertheless, the only physical parameters required for the actual calculations are as before a , y, and V,, as a function of T for each species plus the c ' parameter. ( U T , L'Z, and p are deduced from the molar volumes if the molar vlume a t 0°K is known.) The Flory-Huggins x parameter appears therefore as a recipient where several items of various kinds have been enclosed. The present analysis shows that x is the sum of two entropic and of four enthalpic contributions. Let us recall that the x parameter in Flory's early work was just that included in the term AW/RTpzZ.
Extrapolation of
x
to Zero Polymer Concentration A
necessary prerequisite for eq 38 and 40 to be meaningful in that they should extrapolate to some constant value when pz tends to zero.17 In other words, the term in (p2 should vanish when expanding in powers of pz. This is indeed found to be the case, as shown in the microfilm edition of this paper (see footnote 21).l8 V. Modifications Arising from Unequal Free Volume Fractions of Mixing All the previous results have been derived on the basis of the validity of eq 7. Hypothesis 7 (i.e., the two eq 7) is the simplest possible one and one must start with it. However, it is not the most reasonable. In physical terms, exact validity of this hypothesis would mean that the kinetic energy per molecule or segment is several orders of magnitude greater than the expansion energy per molecule or segment. Simple qualitative calculations show however that the kinetic energy, although greater than, is generally of the same order of magnitude as the expansion energy. Therefore eq 7 will often be an inadequate approximation and the heat of mixing and all subsequent quantities will have t o be recalculated according to inequalities 10. Thus, the two integrals (13a) and (13b) will be replaced by
s,,
To+AT~'
I,'
=
Wi(T) dT
The Journai of Physical Chemistry, Vol. 77, No. 25, 1973
(13a')
K 2 L0
W d T ) dT
=
K21z (13b')
This permits writing the enthalpy of mixing as
A H = nlKiI1
+ n2K212+ nlpz13+ (n, + n J I ,
(28')
If it is assumed, as a first step, that K1 and Kz are nearly independent of composition ( i e . , dKl/dnl = dK1/ dn2 = dKz/dnl = dK2/dn2 = 0), then the equations obtained in paragraphs I11 and IV for the partial enthalpies of mixing and for xH will be very simply modified by multiplying the relevant terms by K1 and K2.
VI. Applications The previous results have been applied (A) to mixtures of n-alkanes and (R) to solutions of polyisobutylene (PIB) in n-alkanes. Since the pair interchange energy is not known exactly and further depends on the volume, the excess enthalpy HE cannot be directly determined using eq 28. Therefore we shall reverse the argument and calculate the quantity 2 = HE - S - 1 4 where H E is the experimental heat of mixing data, and S and 1, are the contributions due to steps a and c of the mixing process. In particular if the value calculated for S is relevant, one should find for Z a value which depends on the temperature but remains positive, as should the pair interchange energy. A. Mixtures of n-Alkanes. The following mixtures of nalkanes have been investigated: c8-c16, C6-cl6, C 6 - c 2 Z 1 and CS-c36. For these mixtures reliable a , y,and vbldata are available.6 However, excess enthalpies H E and excess volumes VE as a function of temperature are only available for the two first mixtures, so that a comprehensive analysis of the two last mixtures is impeded by the lack of sufficient data. The volume a t O"K, which is needed to calculate U T , L'Z, and v m in eq 15 and 16 has been calculated using the empirical relationship established by Do01ittle.l~ The procedure used to calculate the integrals 11 and IZ (eq 13) was the following. First, an approximate estimation of the boundary values AT1 and aT2, as given by eq 15 and 16, was made using the values al(T0) and az(To) a t the mixing temperature for the coefficient of thermal expansion, instead of the mean values 6 and G. Next, improved values ATl' and ATz' were calculated, using in
2981
Derivation of tho Thermodynamics of Polymer Solutions
TABLE I to c6 C8
c16 c22 c36
PIB
1.6053 1.7853 3.0374 3.8624 6.3970 75.031
t’
t*
-6.2060 X -3.9552 X -1.3350 X - 1.3269 X -2.7142 X -2.5837 X
lop3 lo-’ lo-’ lo-’
t3
-3.3765 2.1456 -1.1106 4.7385 -4.1826 4.0139
6.2389 X -1.0321 X 5.5363 X 2.2994 X 5.7134 X 3.5694 X
t4
8.0658 X -8.3747 X 4.4749 x -3.9847 X -7.5212 X 1.9757 X
X X lo-’ X X X X
lo-’’ lo-’’ 10- 1 2 lo-” lo-’’ lo-’’
S Joules)
200 80
150 64
48
100
32
50
16 I
I
1
I
20
40
60
80
I
_
100 t ” C
+
Figure 2. Sum S = x l l (1 - x)12 for Ca-Cj6 mixtures at various volume fractions of c16 as a function of temperature. Figure 1. AT1 and AT2 values vs. volume fraction of c16 for mixtures. The mixing temperatures are indicated on the curves. Curves with increasing slopes refer to ATl.
c8-c16
eq 15 and 16 the values al(T0 + AT1/2) and az(T0 + AT@). Finally, fi and were approximated by the values a1(T0 + AT1’/2) and az(T0 + ATz’/Z). This approximation has proven to be sufficient in all cases excepting C G - C ~mixtures, ~ where the magnitude of the limit AT1 for hexane made necessary a correction resulting from the curvature of the a us. T curve. In all cases the N value used is estimated to approach the true value 5 to better than 0.5%. Once the limits of integration were determined, the polynomials W ( T ) = N ( T ) Y(T)VM ( T ) T were calculated, using the polynomials for a , y, and VM measured by Orwoll and FlorylG (in preference to polynomials of other authors reported in the same work). The expansion polynomials for W(T ) / T are tabulated in Table I, t being the temperature in degrees centigrade. Finally, the compression integrals 1 4 at various temperatures were calculated, using experimental values for the excess volumes and eq 25 to 27. All the above calculations were performed on a Univac 1108 computer. The results for each of the above mixtures will now be briefly discussed. Ca-C16 Mixtures. This mixture approaches a mixture of simple liquids. In Figure 1 the AT1 and AT2 values are plotted and in Figure 2 the sum S = xI1 (1 - r)Zz where x represents the molar fraction of the first component is
+
i km3)
2.c
(cal
40
20
0
- 20
I
1
I
I
20
40
60
I
80
I
100 t’C
Figure 3. Excess volumes VE and excess enthalpies HE for C8and c6-cl6 mixtures: A , VE for c8-c76; 0 , VE for c6-cI6 mixtures; HE for C8-c16; and 0 ,HE for c6-c16 mixtures. From data reported by Orwoll and Flory.6 c16
+,
The Journal of Physical Chemistry, Vol. 77, No. 25, 7973
2982
Jean Dayantis
-501
20
I
I
I
40
60
80
Figure 4. Compression integrals mixtures vs. temperature.
/4
for
CB-c16
I
100 t'C
and
c6-c16
plotted. It is seen that the limits of integration vary almost linearly with the volume fraction of the first component and that the sum S increases regularly with temperature. In Figure 3 the excess enthalpy HE and the excess volume VE are plotted, using data reported in ref 6. The compression integral 1 4 from 10 to 100" is plotted in Figure 4. The important point, using these data, is to determine whether or not the simplest assumption, expressed in the two eq 7, is a convenient approximation. If it is, the molar sum Z = HE - S - 14 should be nearly constant with temperature and equal to AQlnterchenergy. Otherwise equalities 7 should be replaced by inequalities 10 which will lead to smaller values for the limits of integration. The result is shown in Figure 5 . It is seen that 2 is small and positive at 20" and drops to negative values between 60 and 80". Thus, the simplest assumption expressed in eq 7 is not really satisfactory even in the case of CS-c16 mixtures. However the fact that small and positive values are obtained, as expected, for 2 at room temperature is a result favoring eq 13a and 13b. As the temperature is increased, the "mixing" free volume fractions u1m and L ' Z ~ become respectively greater and smaller than u m given by eq 4. In other words, the ratio G/cl,which, for small enough values of u1 and uz (i.e. low enough temperatures), is approximately one, decreases gradually as the temperature is increased. A more quantitative analysis of this effect is not possible for this mixture, since excess enthalpy data above 50" are lacking. c6-cI6 Mixtures. Experimental HE and VE data are plotted in Figure 3, the compression integral 1 4 in Figure 4, and the final result Z in Figure 5 . It is seen that Z presents the same behavior as for cg-c16 measures, but the effect is even more pronounced. Thus Z drops from about +80 J/mol at 20" to -70 J/mol a t 100". To explain this behavior the same argument as for Cg-c16 mixtures can be developed. Abandoning hypothesis 7 we may seek the values u l m and uzm of the mixing free volume fractions, obeying inequalities 10b and 1Oc which will make Z conThe Journal of Physical Chemistry, Vol. 77, No. 25, 1973
't,
Figure 5. Sum Z = HE - S - 14 for c&16 (curve I ) and C6-CI6 (curve I I ) mixtures vs. temperature.
+
stant by conveniently reducing the sum S = ~ 1 1 (1 x ) I z . A detailed calculation along these lines is included in the microfilm edition of this paper (see footnote 22).18 Let us only point out that it is always possible to choose u l m and uzm, linked by one relationship, so as to keep constant the value of Z when the temperature varies. The exact form of the relationship which links u l m and u p is not known, however, the only condition imposed on it, i.e., an excess volume of mixing equal to zero, being insufficient to define it unambiguously (see microfilm edition, footnote 23).18 C6-cZ2 Mixtures. HE and VE data are not available for this mixture. Comparison with HE and VE data for c6-c24 mixtures6 suggests, at 50°, a positive and small value for 2. CS-c36 Mixtures. This mixture is nearly a polymer-solvent system. At 80" and for the equimolar volume fraction of c36 (i.e., (a2 = 0.82), the sum S = ~ 1 1 (1 - X ) ~ Zis equal to 710 J/mol. At the same temperature the com111 pression integral 1 4 is equal to -430 J/mol and HE J/mol, so that 2 = -170 J/mol. This substantially negative value for Z should be ascribed to rather severe departure from hypothesis 7. A more comprehensive study of this mixture is not possible at present since HE and VE data are lacking for temperatures other than 76". In Figure 6 are plotted the sums S for the four previous mixtures, reported to 1 g of mixture at 80". The curves are almost symetrical about the (FZ = l/z axis. B. Solutions of Polyisobutylene in n-Alkanes. The following mixtures have been considered: C6-PIB, Cs-PIB, and C16-PIB. In fact, as a result of the use of a computer, the only limitation in studying more systems was imposed by the availability of proper expansion polynomials for a , y, and For example, use of mean values between c6 and C S for the expansion polynomial y of C7 has lead to erroneous results, so that the study of the system CT-PIB had to be discarded. The expansion polynomials used for PIB were those determined by Eichinger and Flory.8a Excess enthalpies at 25" for the above systems were taken from the paper by
+
v,.
-
2983
Derivation of t h e Thermodynamics of Polymer Solutions
TABLE II
Vol fraction of polymer
Vol of polymer/mole mixture at 25" om3
Wt of polymer/mole mixture, g
No. of mole equiv of C4Hs
HE/mole C4HaSa J
HE/mole mixture, J
C6-PIB c8-PIB
0.540 0.532
154.3 182.8
141.5 167.6
0.515
303.5
278.3
-142.1 -66.9 4-2.5
- 165.2
C16-PIB
2.527 2.993 4.970
Mixture
VE,
S,J
-2.44 - 1.65 -0.47
572.5 414.4 103.5
CG-PIB c8-PIB cj 6-PI B
cm3/mole mixture
i 4 , J/mole mixture
Z,J/moie
-808.5 -552.4 -162.3
+70.8 4-45.1 +64.9
-93.7 4-6.1
mixture
Reference 9
PIB, one has to multiply by the number of moles equivalent of C4Hs and the volume fraction of the solvent. This is done in the seventh column. The final result reported in the last column shows that 2 is remarkably constant with the chain length of the n-alkane considered. It is very unfortunate that HE and VE data are not available at higher temperatures, to follow, as for n-alkane mixtures, the variation of Z with temperature.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
'f2-
Figure 6. S u m S divided by the molecular weight of the mixture for C8-CI6, C6-cJ6, &-C22, and cs-c36 mixtures at 80" as a function of the volume fraction of the component of higher molecular weight.
Delmas, et ~ l . and , ~ excess volumes from that by Flory, et aL7 The volume at 0°K of PIB, required to determine free volume fractions, was estimated from the expansion volume of PIB at the glass transition temperature given by BondPo (i.e. 0.125 a t -70", see also ref 15). All calculations were performed considering a PIB of molecular weight of 10.000, with an estimated molar volume at 0°K of 9.059 cm3. Molar fractions have been calculated as a function of volume fractions on this basis. Data for n-alkanes were the same as previously. Results for S = xI1 (1 - x ) Z p between 20 and loo", assuming validity of hypothesis 7 are given in the microfilm edition of this work.lB Excess enthalpies of mixing H E , excess volumes VE, compressioii integrals 14, and final result Z = HE - S - 1 4 at 20" are summarized in Table 11. In the sixth column of Table I1 are given the enthalpies of mixing per mol of C4Hg a t infinite dilution, as reported by Delmas, et aL9 To determine the excess enthalpy of a given weight of
+
VII. Conclusions Phenomenological thermodynamics has been applied to the problem of the heat of mixing in solutions. No structure whatever has been assumed for the pure components in the liquid state and the solutions. Application of the theory to mixtures of n-alkanes and to mixtures of polyisobutylene with n-alkanes, after making the additional hypothesis that the mixing free volume fractions of the components are equal, leads a t room temperature to small and positive values for the interchange energy. (The only heat effect considered in conventional theories of solutions.) This is a satisfactory result, the more so since the integrals 11 and 1 2 range from several thousands to several tens of thousands Joules per mole. However, for c8-c16 and CS-ClS mixtures, the sum 2 = HL - ( x I 1 + (1 - x ) l z ) - 1 4 becomes negative as the temperature is increased from 20 to loo", and is already negative for c6-c36 mixtures a t 80". This behavior has been ascribed to the fact that the hypothesis of equal mixing free volume fractions in the components before mixing becomes less and less valid as the temperature is increased. Unfortunately, since HE and VE data as a function of temperature are not available at present, it is not possible to follow the variation of 2 with temperature for PIB-n-alkane mixtures also. What the value of the mixing free volume fractions, which need to be considered in the general case, is still unresolved. An attempt to treat this problem was made for c6-c16 mixtures (see the microfilm edition). It seems however that a more rigorous procedure would require the use of three pair distribution functions, one for each kind of pair. Acknowledgments. The author expresses his recognition to Professor H. Benoit, Director, Centre de Recherches sur les Macromolhcules, Strasbourg, for his constant support during the course of this work. He also thanks Dr. C. Guez of the "Centre de Calcul de Strasbourg-Cronenbourg" for helpful advice in writing the computer program. The Journai ot Physicai Chemistry, Voi. 77, No. 25, 1973
2984
E.
Supplementary Material Auailable. Footnotes 21-23 and Figures 7-16 will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 20X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $4.00 for photocopy or $2.00 for microfiche, referring to code number J P C -73-2977. efeesences and Notes (1)
J . Dayantis, J . Phys. Chem., 76, 400 (1972). See also C. t?. Acad. Sci., Ser. C, 275, 1254 (1972).
( 2 ) I. Prigogine (with the collaboration of A. Beilemans and V. Mathot), "The Molecular Theorie of Solutions," North-Holiand Publishing Co.. Amsterdam, 1957. (3) (a) P. J. Flory, R. A. Orwoll, and A. Vrij, J . Amer. Chem. Soc., 86, 3507 (1964); (b) 86, 3515 (1964). (4) P. J. Flory, J. Amer. Chem. Soc., 87, 1833 (1965). (5) A. Abe and P. J. Flory, J. Amer. Chem. Soc., 87, 1838 (1965). (6) (a) R. A. Orwoll and P. J. Flory, J. Amer, Chem. Soc., 89, 6814 (1967); (b) 89, 6822 (1967).
(7)
(8)
(9) (10) (11) (12) (13)
(14) (15) (16) (17)
(18)
(19) (20)
McLaughlin, L. H. Hall, and R . W. Rozett
P. J. Flory, J. L. Ellenson, and B. E. Eichinger. Macromolecules, 1 , 279 (1968). (a) B. E. Eichinger and P. J. Flory, Trans. Faraday Soc., 64, 2035 (1968); (b) 64, 2053 (1968); (c) 64, 2061 (1968); (d) 64, 2066 (1968): (e) Macromolecules, 1, 285 (1968). G. Delmas, D. Patterson, and T. Somoynsky, J. Polym. Sci., 57, 79 (1962). D. Patterson, J. Polym. Sci., Part C, 16, 3379 (1968). J. A. Barker, "Lattice Theories of the Liquid State," Pergamon Press, Elmsford, N.Y., 1963. J. M. H. Levelt and E. C. D. Cohen in "Studies in Statistical Mechanics," J. de Boer and G. E. Uhlenbeck, Ed., North-Holland Publishing Co., Amsterdam, 1964. R. N. Haward, J. Macrornol. Sci., Rev. Macromol. Chem., C(4), 191 (1970). The reader will find in this paper a detailed account of the different methods which permit the determination of the volume at 0°K. R.L. Scott, J. Phys. Chem., 64, 1241 (1960). M . L. McGlashan, K. W. Morcom, and A. G . Williamson, Trans. Faraday Soc., 57, 601 (1961). E. A. Guggenheim, "Mixtures," Clarendon Press, Oxford, 1952. Since dilute solutions ( ( ~ 2 < 2-3%)) are not considered in the present treatment, extrapolation.of x to zero concentration is somewhat arbitrary. We may, however, admit for present purposes that there is no sharp variation of the x vs. (FZ curve when ( F Z tends to zero, so that the values obtained by extrapolation of eq 38 and 40 remain meaningful. See paragraph at end of paper regarding supplementary material. A. Doolittle, J. Appl. Phys., 22, 1471 (1951). A. Bondi, J. Polym. Sci., PartA-2, 2, 3159 (1964).
Monoisotopic Mass Spectra of Some Boranes and Borane Derivatives' E. McLaughlin, L. H. Hall,z and R. W. RozettX Department of Chemistry, Fordham University, Bronx, New York 10458 (Received July 5, 1973) Publication costs assisted by the Petroleum Research Fund
Least-squares-fitted monoisotopic mass spectra for BjHsBr, C2&-&0&, B4H10, &+ill, and &OH16 are given. Each represents the best monoisotopic mass spectrum presently available from measurements with a conventional mass spectrometer. Isotope cluster analysis is also used to show that the spectrum reported as &OH26 results from a mixture of CgBloHzs and probably CgBsH29.
Introduction restrict the intensities to nonnegative values simply by removing the formula of the peak in question, their presence Analysis of the cluster of intensities due to the isotopic is always a sign of badly fitting elemental formulas.6 A variants of the elemental formulas found in a mass specwell-fitting spectrum will have no negative intensities; a trum allows one to simplify spectra, detect impurities, completely inappropriate set of formulas will produce on and establish the elemental composition of the ions if the average a sum of negative intensities equal to the sum strongly polyisotopic elements are The proceof the positive intensities in the monoisotopic spectrum. dure has become convenient with the development of a computer program called MIMS which handles any numMonoisotopic Mass Spectra ber of overlapping isotope clusters, any combination of elements, and any fractional abundance of their i ~ o t o p e s . ~ Monoisotopic mass spectra prepared with the program MIMS are listed in Tables I-IV. Unpublished digital mass MIMS uses all the polyisotopic information available to spectra of the halogenated pentaboranes were used to gengenerate a least-squares-fit monoisotopic mass spectrum erate the monoisotopic spectra of Tables I and II.7,8 The with intensities restricted to positive values. Two measpectrum of ethyldecaborane (Table 111) is a much imsures of success are available from the program. The first proved calculation from published polyisotopic intensiis the root-mean-square deviation, RMD, the usual meaties.g The mass spectra of B4H10 and B5Hll have been sure of fit of a least-squares procedure. Past experience studied by many worker^.*,^^-^^ The intensities of Tables suggests that the present method provides a fit 25 times IV and V represent the best-fitting results using data from smaller on the average than hand-fitted spectra.6 The seca conventional mass spectrometer.? &OH16 was reinvestiond criterion is the absence of negative intensities in the gated because of its poor fit (RMD 1.8).6 The original digmonoisotopic spectrum. While the program allows one to The Journai of Physical Chemistry, Vol. 77, No. 25, 1973
